How Vertex Algebras Are Almost Algebras: on a Theorem of Borcherds

HOW VERTEX ALGEBRAS ARE ALMOST ALGEBRAS: ON A THEOREM OF BORCHERDS

CONTENTS

1. Introduction 1 2. Algebraic Preliminaries 4 3. Vertex Algebras 5 4. Borcherds’s theorem 6 5. Converse to Borcherds’s theorem 8 6. Examples 10 7. Some Questions 12 References 12

1. INTRODUCTION

1.1. Today, vertex algebras have become rather ubiquitous in mathematics. Among the subjects on which they bear direct influence, we can list: finite group theory, number theory (both through the classical theory of elliptic functions and the modern geometric Langlands theory), combinatorics, representation theory, and algebraic geometry. Before we discuss these relations, let us recall that despite their firm entrenchment within the world of pure mathematics, vertex algerbas (or rather their consituent elements vertex operators) actually first arose in the 1970s within the early string theory literature concerning dual resonance models [ref?]. Independently of this, [LW] Lepowsky and Wilson were interested in constructing representations of the affine Lie algebra slb2 using differential operators acting on the ring of infinite polynomials C[x1,x2,...]. The main difficulty in doing so was to construct representations of a certain infinite Heinsenberg algebra using rather complicated looking formulae. Howard Garland then noticed that these same formulae occurred within the string theory literature [S], and he christened the term "vertex operator" to the mathematical world. Shortly thereafter, Igor Frenkel and Victor Kac (and G. Segal independetly) made precise sense of Garland’s observation and constructed the most important representation of arbitrary affine Lie algebras, the so called basic representation, again using vertex operators to realize certain infinite Heinsenberg subalgebras. Thus within the world of representation theory of affine Lie algebras, vertex operators play a central role. What may come as a surprise is how pervasive vertex algebras have now become in mathematics. The first dramatic use of vertex operators is in the theory of finite groups. We refer to the excellent book of Frenkel-Lepowsky-Meurman [FLM] for more details, but here is a sketch. In the classification of finite groups, the Fischer-Greiss Monster (aka, friendly giant) is the largest of the 26 sporadic simple groups. A number of very mysterious empirical phenomena concerning this group began to surface in the papers [? , ?, ?] These results collectively went under the title Monstrous moonshine, and one particular facet of moonshine was the prediction was that most natural representation of this finite simple group was actually of infinite dimension!. The question then arose as to how to realize this representation. Again, vertex operators proved to be essential here: in [FLM?], the moonshine module, the desired infinite dimensional representation of the Monster group, was constructed with the aid of vertex operators. R. Borcherds then proposed an axiomatic framework to deal systematically with vertex operators, the so called vertex operator

Date: December 11, 2007.

1 algebras. In [FLM], the moonshine module was then naturally given a vertex operator algebra structure and moreover it was shown that the Monster could be realized as the group of automorphisms of this vertex operator algebra. Motivated by these results, it was well expected that other vertex operator algebras should similarly encapsulate the symmetries of the other sporadic simple groups. However, it was not until the work of John Duncan [ref?], nearly two decades after the pioneering work on FLM, that this old dream was to come to fruition: in [], Duncan exhibits constructions of other sporadic simple group (even some which are not contained within the Monster!) using some generalizations of vertex operators algebras. Already through the construction of the Moonshine module, vertex operators were known to have number theoretic significance. Indeed, the graded dimension of the Moonshine module is the elliptic function j(τ). A deep generalizatin of this results, Yongchang Zhu showed the modular invariance of the characters of modules over vertex algerbas. A more recentl connection to number theory stems from Beilinson and Drinfeld’s geometric reformulation of the axioms of vertex algebras as chiral algebras. Using these chiral algebras in an essential way, Beilsion and Drinfeld are able to prove certain instances of the geometric Langlands correspondence. The importance of vertex operators in combinatorics stems from the realization of the basic representation of affine Lie algebras in the space of symmetric polynomials in infinitely many variables. The combinatorics of vertex operators are then reflected in the properties of symmetric functions. For example, the Boson- Fermion correspondence from physics has been translated by I. Frenkel into an explicit isomorphism of vertex algerbas. This isomorhism in turn encodes (or is encoded) by the Littlewood-Richardson coefficients, a basic object in any combinatorialists tool kit. This paper was motivated by a connection between vertex algebras and the geometry of algebraic surfaces. The work of Nakajima and Grojnowski [ref?] exhibited a remarkable geometric construction of the Heisenberg algerba on the cohomology of Hilbert schemes of points on a surface. Moreover, certain very special vertex opertors were also shown to have geometric significance in terms of Chern classes of tauto- logical bundles on the Hilbert scheme [ref?]. We may then pose the question: what is the geometric meaning of more general vertex operators? The answer to this question still remains quite mysterious. Our approach was a synthetic one which attempted to first pare down the axioms of a vertex algebra so that each part may have a clear geometric meaning.

1.2. Having commented on the central place of vertex algebras within mathematics, one might naturally wonder: what exactly is a vertex algebra? Unfortunately, this question does not have a very short answer. For example, the statement that a vertex algerba is a particular type of algera is false! It would be more accurate to say that a vertex algebra is a vector space endowed with an inﬁnite family of compatible multi- plications. To make sense of this statement, one usually adopts the axiomatic framework of B, FLH. From this point of view, the basic data of vertex algebra is a pair (V,Y(·,z)) where V is a vector space and Y is a map, Y(·,z) : V → End(V)[[z±1]].

In other words, to each v ∈ V, we attach an infinite number of endomorphisms of V, say v(n). Thus, given two elements v,w ∈ V, we have an infinite collection of products v(n)w ∈ V. The axioms for a vertex algerba are then ... as having an infinite number of algebra structures. There is a very clear axiomatic characterization of vertex algerbas which begins [B] and is fully carried out in [FLH]. Recently, partly motivated by a desire to q-deform vertex algebras, Borcherds has again taken up basic question as to what exactly is a vertex algebra. This note then aims to provide an exposition of one aspect of R. Borcherds’s paper "Quantum Vertex Algebras." The reader, though, will find no trace of any sort of "quantum" deformation in this paper. Rather, we will be concered here with a simple new construction of a certain class of vertex algebras (including Heisenberg and Lattice vertex algebras) described in [?]. Starting from some rather innocuous algebraic elements– a commutative, cocommutative bialgebra V equipped with a derivation T : V → V and a symmet- − ric bicharacter r : V ⊗V → C[(z − w) 1]– Borcherds shows how to construct the richer structure of a vertex algebra on V.

2 In this formulation, a natural predecessor to Borcherds’s result is the simple fact (due to Borcherds too?) that a holonomic vertex algebra structure on V, i.e., one in which the fields Y(a,z)v ∈ End(V)[[z]] are just power series, is equivalent to the structure of a commutative, unital algebra with derivation on V. Briefly, this equivalence is constructed as follows: given a holonomic vertex algebra, we may define a product on V by a · b := [Y(a,z)b]z0 where by [A(z)]z0 denotes the coefficient of the constant term of A(z) ∈ V((z)). With this definition, the vertex algebra axioms imply that V becomes a commutative unital algebra with |0i := 1, and the derivation on V is just the translation operator T from the vertex algebra structure. In the other direction, given a commutative unital algebra with derivation T, we can construct vertex operators by the formula Y(a,z)b = ezT (a)·b. The axioms for a vertex algebra then follow readily from the ring properties V. Since the most interesting vertex algebras are not holonomic, we might wonder how to modify the above construction by somehow introducing singularities or non-trivial OPEs into the picture. Borcherds provides a partial answer this question as follows: given a bialgebra V, a derivation T compatible with the bialgebra − structure, and bicharacter r :V ⊗V → C[(z−w) 1] he constructs a vertex algebra on V with 2-point functions (which govern the most singular part of the OPE) specified by r. The vertex operators are then given by the following elegant formula which combines all the data which we are given:

Y(a,z)b = ∑T (k)(a0) · b0 · r(a00,b00), (a)

0 00 where the coproduct on V is written as ∆(a) = ∑(a) a ⊗ a . Our first goal will be to prove that this formula does in fact define a vertex algebra on V. We refer the reader who may be wondering how such a formula was arrived at to [?], where the above formula is naturally fit into the framework of twisted group rings. Our point of view will be to take such a formula as a starting point and directly show that it satisfies the axioms of a vertex algebra. This is slightly different from Borcherds’s original proof which sets up a very general framework in which the desired result follows quite formally. This framework requires a healthy dose of abstract nonsense which might be less than transparent at first glance, and so we hope this note will be a servicable introduction to [?]. The next question we address is what types vertex algebras can be constructed using the new approach. Along these lines, we define a certain class, called r-vertex algebras, which contain the vertex algebras constructed as above from bialgebras V. For example, Heisenberg and Lattice vertex algebras are contained in this class. The second main result in this note is a uniqueness theorem which shows that if the underlying bialgebra of an r-vertex algebra V is primitively generated, then it is the vertex algebra as constructed above from a bialgebra V, derivation T, and bicharacter r determined by the 2-point functions. We do not know whether this uniqueness result can be extended to a larger class of r-vertex algebras, but we make some comments about this problem in the final section. We would like to mention here that our motivation seems quite different from Borcherds’s. We refer to [?] for a more details, but very briefly, Borcherds’s construction, inspired in part by physics, seems to have been designed so as to be naturally modifiable to produce examples of what he terms "quantum" vertex algebras. Our motivation, on the other hand, comes from a natural attempt to "geometrize" vertex algebras. The bosonic Fock space representation of the Heisenberg algebra admits several beautiful geometric descriptions in terms of Hilbert schemes of points on surfaces [?], infinite grassmannians [], or equivariant K-theory. This Fock space also carries the structure of a Heisenberg vertex algebra, so it is a natural, if somewhat vague question, to determine what is the geometric meaning of the entire vertex algebra. Using the results in this note, we may say that a vertex algebra is nothing more than a bialgebra with derivation together with two- point functions. Furthermore, since both the bialgebra and derivation have been given geometric meanings, the main obstacle to ’geometrizing’ vertex algebras seems to be to find out the geometric significance of the 2-point function r. Finally, this paper is organized as follows: In section 2, we set up some preliminary algebraic machinery. The reader is advised to atleast skim over this part as many terms like bialgebra, derivation, etc. are given slightly nonstandard meanings. In section 3, we review some basics from the theory of vertex algebras, and

3 in section 4, we state and prove Borcherds’s theorem. In section 5, we state a converse, and we conclude with some examples in section 6 and some questions in section 7. Acknowledgements: This paper grew out of discussions with Igor Frenkel during the Spring 2003 semes- ter. The main observation, that Borcherd’s bicharacter r(·,·) was merely a 2-point function, is due to him. I would like to thank him for his encouragement and many helpful suggestions on this manuscript. Also, I would like to thank Minxian Zhu and John Duncan for many tutorials on vertex algebras. This research was conducted while the author was supported by an NSF graduate research fellowship.

2. ALGEBRAIC PRELIMINARIES

We ﬁx an algebraically closed ﬁeld k of characteristic zero.

2.1. Algebras, Coalgebras, and Bialgebras. We take an algebra A will be a commutative Z+-graded ∞ k-algebra with unit, A = ⊕n=0An, where each An is a ﬁnite dimensional k-vector space. ∞ Dually, a coalgebra will be a cocommutative Z+-graded k-coalgebra (C,∆,ε) where if C = ⊕n=0Cn, then n Cn is a ﬁnite k-vector space, ε(Cn) = 0 for n 6= 0, and ∆Cn ⊂ ∑i=0 Ci ⊗Cn−i. We say that a coalgebra is connected if C0 = k. 0 00 Remarks 2.2. We adopt Sweedler notation to write the coproduct, ∆(x) = ∑(x) x ⊗ x . Generalizing, we will also write for example (1) (2) (3) ∆2(x) = ∆ ⊗ 1)∆(x) = (1 ⊗ ∆)∆(x) = ∑ x ⊗ x ⊗ x . ∆2(x)

We deﬁne a bialgebra to be an algebra A with a coalgebra structure (A,∆,ε) such that the gradings on A as an algebra and coalgebra agree and ∆ and ε are algebra morphisms.

2.3. Bialgebras with Derivation. By a bialgebra with derivation, we shall mean bialgebra V with a linear map T : V → V be a degree 1 satisfying (1) T(1) = 0 (2) T(a · b) = T(a) · b + a · T(b) 0 00 0 00 (3) T(∆(a)) = ∑(a) T(a ) ⊗ a + ∑(a) a ⊗ T(a )

(k) We say that a set {aα |aα ∈ V}α∈S is a T-generating set of an algebra V if {T aα }α∈S generates V as (k) T k an algebra. Here, as usual, T := k!

2.4. Bicharacters. We next describe the additional piece of information on a bialgebra with derivation which allows us to construct interesting vertex algebras.

Deﬁnition 2.5. Let V be a bialgebra with derivation T. Then a bicharacter is a bilinear map rz,w : V ×V → k[(z − w)±1] satisfying (1) r(a,b) = r(b,a); (2) r(a,1) = ε(a); 0 00 (3) r(a · b,c) = ∑c r(a,c )r(b,c ) d (4) r(Ta,b) = − dz r(a,b) d (5) r(a,Tb) = dw r(a,b). c(a,b) (6) For a ∈ Vm and b ∈ Vn, r(a,b) = (z−w)−m−n , where c(c,b) is some scalar. Suppose now (V,T,r) is a bialgebra with derivation and bicharacter. In what follows, we will also make use of the following multivariable generalization of r.

4 Deﬁnition 2.6. Let a1,...,an ∈ V. Then we recursively deﬁne ±1 rz1,...,zn : V ×V ···V → k[(zi − z j) ]1≤i6= j≤n, to be r (a ,...,a ) = r (a0 ,...,a0 ) n r (a( j−1),a00,...,a00,...a00) z1,...,zn 1 n ∑ z2,...,zn 2 n Π j=2 z1,...,zˆj,...zn 1 2 bj n ∆n−1(a1),(a2),...,(an) Example 2.7. To make things more transparent, we unravel the above formula to get, 0 0 0 00 00 00 rx,y,z(a,b,c) = ∑ ry,z(b ,c )rx,y(a ,b )rx,z(a ,c ) (a),(b),(c) and similarly

(1) (1) (2) (1) (3) (1) (2) (2) (3) (2) (3) (3) rx,y,z,w(a,b,c,d) = ∑ rx,y(a ,b )rx,z(a ,c )rx,w(a ,d )ry,z(b ,c )ry,w(b ,d )rz,w(c ,d ) ∆2(a),∆2(b),∆2(c),∆2(d)

Using the properties of r and the (co)commutativity of V, it should come as no surprise that

Lemma 2.8. Let a1,...,an ∈ V and σ ∈ Sn a permutation. Then

rz1,...,zn (a1,...,an) = rzσ(1),...,zσ(n) (aσ(1),...,aσ(n)).

∨ ∞ ∗ 2.9. Bilinear forms. Given a bialgebra V, we can form its graded dual V = ⊕n=0Vn . By a bilinear form on V we will mean the following a non-degenerate, symmetric, bilinear form (·,·) on the vector space V satisfying ∨ (1) (Vm,Vn) = 0 unless m = n (identifying V with V naturally) (2) (1,v) = ε(v)

3. VERTEX ALGEBRAS

We must first introduce the concept of a field. Definition 3.1. Let V be a vector space over k. A formal power series − j ±1 A(z) = ∑ A( j)z ∈ EndV[[z ]] j∈Z is called a field on V if for any v ∈ V we have A( j) · v = 0 for large enough j.

In case V is Z+ graded, we can define a field of conformal dimension d ∈ Z to be a field where each A( j) is a homogeneous linear operator of degree − j + d

A useful procedure for dealing with ﬁelds will be normal ordering, deﬁned as follows.

Definition 3.2. Let Let A(z),B(w) be fields, and denote by A(z)+ and A(z) the non-negative and negative parts (in powers of z) of A(z). Then define : A(z)B(w) := A(z)+B(w) + B(w)A(z)− to be normally ordered product of A(z) and B(w).

We are now in a position to define the main object of study, Definition 3.3. A vertex algebra (V,|0i,T,Y) is a collection of data: ∞ • (state space) a Z+ graded k-vector space V = ⊕m=0Vm, with each dimVm < ∞ • (vacuum vector) |0i ∈ V0 • (translation operator) T : V → V of degree 1. ±1 • (vertex operators) Y(·,z) : V → End(V)[[z ]] taking each A ∈ Vm to a field −n−1 Y(A,z) = ∑ A(n)z n∈Z of conformal dimension m (i.e., degA(n) = −n + m − 1).

5 satisfying the following axioms:

• (vacuum axiom) Y(|0i,z) = IdV . Furthermore, for any A ∈ V, Y(A,z)|0i ∈ V[[z]] and Y(A,z)|0i|z=0 = A. d • (translation axiom) For every A ∈ V, [T,Y(A,z)] = dzY(A,z) and T|0i = 0 • (locality 1) For every A,B ∈ V, there exists a positive integer N such that (z−w)N[Y(A,z),Y(B,w)] = 0 as formal power series in End(V)[[z±1,w±1]].

Corellation Functions. Let (V,|0i,T,Y) be a vertex algebra, and assume that it is equipped with a non- degenerate, symmetric, bilinear form (·,·) which identiﬁes V with its restricted dual V ∨. With this assumption, denote h0| ∈ V the dual to the element |0i ∈ V. Then, we introduce the

Deﬁnition 3.4. Given a1,...,an ∈ V. We deﬁne the n-point functions to be

(h0|,Y(a1,z1)···Y(an,zn)|0i) ∈ k[[z1,z2,...,zn]][(zi − z j)]1≤i< j≤n

4. BORCHERDS’STHEOREM

In this section, we state and prove Borcherds’s theorem [?]. Let V be a bialgebra with derivation T and bicharacter r. We aim to construct a vertex algebra on V with translation operator given by V and 2-point functions specified by r. We first define the following maps which will allow us to construct vertex operators, Definition 4.1. Let n −1 Φ (z1,...,zn) : V ⊗V ⊗ ··· ⊗V → V[[z1,z2,··· ,zn]][(zi − z j) ] send ( j1) 0 ( j2) 0 ( jn) 0 00 00 00 a1 ⊗ a2 ···an 7→ ∑ ∑ T (a1) · T (a2)···T (an)rz1,...,zn (a1,a2,...,an). (a1)···(an) j1,... jk≥0

So, for example, we have that 2 ( j) 0 (k) 0 00 00 Φ (z,w)(a,b) = ∑D (a )D (b )rz−w(a ,b ). The following is easy to see from (co)commutativity of V and the corresponding facts for r,

Lemma 4.2. Let a1,...,an ∈ V and σ ∈ Sn a permutation. Then n n Φ (z1,...,zn)(a1,...,an) = Φ (zσ(1),...,zσ(n))(aσ(1),...,aσ(n)) −1 as elements of V[[z1,...,zn]][(zi − z j) ]1≤i< j≤n.

Following Borcherds, we make the following fundamental definition: Definition 4.3. Let a,b ∈ V. Then define the field Y(a,z) by

Y(a,z1)Y(b,z2)|0i = Φ(a,b)(z1,z2).

In particular, we have Y(a,z1)b = Φ(a,b)(z1,0).

With the above deﬁnition, we can verify the following formula for multiplication.

Lemma 4.4. Let a1,...,an ∈ V. Then Y(a1,z1)···Y(an,zn)|0i = Φ(a1,...,an)

Proof. The proof is a straightforward computation. For ease of notation, we just sketch the case n = 3 which already illustrates all the main ideas. Let a,b,c ∈ V. Then from the above deﬁnition, we see that k l (k) 0 (l) 0 00 00 Y(a,z1)Y(b,z2)Y(c,z3)|0i = Y(a,z1) ∑ ∑ z2z3T (b )T (c )rz2,z3 (b ,c ). (b),(c) k,l≥0 Expanding 00 00 (k) 0 (l) 0 k l ∑ rz2,z3 (b ,c ) ∑ Y(a,z1)[T (b )T (c )]z2z3, (b),(c) k,l≥0

6 we get 00 00 k l 00 (k) 0 (l) 0 00 j ( j) 0 ( j) 0 (k) 0 00 ∑ rz2,z3 (b ,c ) ∑ z2z3 ∑rz1 (a ,[T (b )T (c )] ) ∑ z1T (a )[T (b )T (c )] , (b),(c) k,l≥0 (a) j≥0 which using the fact that ∆(a · b) = ∆(a) · ∆(b) and property (3) of r becomes,

00 00 k l (2) ( j) 0 00 (3) (k) 0 00 j ( j) 0 ( j) 0 0 (k) 0 0 ∑ rz2,z3 (b ,c ) ∑ z2z3 ∑rz1 (a ,[T (b )] )rz1 (a ,[T (c )] ) ∑ z1T (a )[T (b )] [T (c )] . (b),(c) k,l≥0 (a) j≥0

Observing that, k ∆T (k)(a) = ∑ ∑D(k− j)(a0) ⊗ D( j)a00, j=0 (a) we collect terms in the above coefﬁcients T ( j)(a(1))T (k)(b(1))T (l)(c(1)) to get

( j) (1) (k) (1) (l) (1) j k+p l+q (2) (p) (2) (2) (q) (2) ∑ ∑ T (a )T (b )T (c ) ∑ z1z2 z3 rz1 (a ,T (b )rz1 (a ,T (c ). j,k,l≥0 (a),(b),(c) p,q≥0 But Taylor’s formula and the property (5) of r give that (n) n rz,w(a,b) = ∑ rz(a,T (b))w . n≥0 Applying this fact to the above sum, we conclude that the terms with coefficient T ( j)(a(1))T (k)(b(1))T (l)(c(1)) 00 00 00 is just rz1,z2,z3 (a ,b ,c ) which concludes the proof. We are now ready to state the first theorem of this note, Theorem 4.5. Let V be a bialgebra with derivation T and bicharacter r. Define fields Y(a,z) as in Definition 4.3. Then (V,|0i,T,Y) is a vertex algebra.

−1 Proof. It is obvious that the map Y(·,z) : V → V[[z]][z ] takes a ∈ Vm to a ﬁeld of conformal dimension m. So, let us verify the vacuum, translation, and locality axioms. ∞ (k) k Vacuum: For a ∈ V, the above deﬁnition gives that Y(a,z)|0i = ∑k=0 T (a)z , from which the vacuum axiom follows easily. Translation: The translation axiom requires that d [T,Y(a,z)]b = Y(a,z)b. dz Expanding the left hand side, we get " # (k) 0 0 00 00 k TY(a,z)b = T ∑ T (a )b rz(a ,b )z (a),(b),k≥0 and (k) 0 0 00 00 k (k) 0 0 00 00 k Y(a,z)Tb = ∑ T (a )T(b )rz(a ,b )z + ∑ T (a )b rz(a ,Tb )z (a),(b),k≥0 (a),(b),k≥0 Since h i T T (k)(a0)b0 = (k + 1)T (k+1)(a0)b0 + T (k)(a0)T(b0), this difference equals (k+1) 0 0 00 00 (k) 0 0 00 00 k ∑ (k + 1)T (a )b rz(a ,b ) − T (a )b rz(a ,Tb )z (a),(b),k≥0 which is easily seen to be equal to

d (k) 0 0 d 00 00 k (k) 0 0 00 00 d k Y(a,z)b = ∑ T (a ) · b [rz(a ,b )]z + T (a ) · b rz(a ,b ) [z ] . dz (a),(b),k≥0 dz dz

7 Locality: Locality essentially follows from the Lemma above. Indeed, let a,b ∈ V. Then Φ3 (a,b,c) = Y(a,z )Y(b,z )c (z1,z2,0) 1 2 and similarly Φ3 (b,a,c) = Y(b,z )Y(a,z )c. (z2,z1,0) 2 1 So, by Lemma Φ, we see that both Φ3 (a,b,c) = Φ3 (b,a,c) ∈ End(V)[[z±1,z±1]][(z − z )−1] (z1,z2,0) (z2,z1,0) 1 2 1 2 which means that there is some positive integer N such that N (z − w) [Y(a,z1),Y(b,z2)]c = 0. Furthermore, this number N does not depend on c as is clear from looking at the explicit formula for Φ. In case our bialgebra V also has a bilinear form, we have the following formula for the n-point functions. Corollary 4.6. Let (V,|0i,T,Y) be the vertex algebra constructed above and let (·,·) be a bilinear form on V. Then

(h0|,Y(a1,z1)···Y(an,zn)|0i) = rz1,z2,...,zn (a1,a2,...,an)

5. CONVERSETO BORCHERDS’STHEOREM

In this previous section, we showed how starting from a bialgebra V together with a derivation T and a bicharacter r we can construct a vertex algebra. Now, given a vertex algebra (V,|0i,T,Y) whose underlying space V also has a bialgebra structure with derivation T, we might wonder to what extent does this completely determine the vertex algebra structure. By itself, V and T cannot know about the singularities (hence, 2-point functions) of Y(·,z), so in order to get a meaningful answer to our question, we have to also feed in the information of r. If the bialgebra structure on V is not reflected in the vertex algebra structure, we will have no hope of recovering the vertex algebra structure. So, in addition to r, we need to assume some compatability conditions between the vertex and bialgebra structures. These considerations are formalized in the following definition. Definition 5.1. An r-vertex algebra consists of the following data: • a vertex algebra (V,|0i,T,Y) • a bialgebra structure with derivation (V,T) with compatible gradings. • a bilinear form (·,·) on V satisfying the following additional axioms:

• for a ∈ V primitive, a(−1)b = a · b

• for a,b ∈ V primitive, a(0)b = 0

With the above deﬁnition, we can state a partial converse to Borcherds’s theorem. This is not the strongest statement that can be made, but the following theorem (or a slight variation of it) sufﬁces to deal with all of the r-vertex algebras which we know. Theorem 5.2. Let V be a connected bialgebra with derivation T, bicharacter r and bilinear form (·,·). Assume that V is T-generated by a set of primitive elements {aα }α∈S. Then there exists a unique r-vertex algebra structure on V such that the 2-point functions of V are given by r.

Proof. In the previous section, we showed how to construct a vertex algebra on V with two-point functions speciﬁed by r. It is clear that this vertex algebra is actually a r-vertex algebra. To prove the uniqueness, we shall make us of the following.

8 ∞ Theorem 5.3. Reconstruction Theorem Let V = ⊕n=0Vn be a Z+-graded vector space, |0i ∈ V0 a non- α zero vector, and T a degree 1 endomorphism of V. Let S be a (countable) set and {a }α∈S a collection of homogenous vectors in V. Suppose we are given ﬁelds α α −n−1 a (z) = ∑ a(n)z n∈Z such that the following conditions hold:

(1) For all α, aα (z)|0i = aα + z(···) α α (2) T|0i = 0 and [T,a (z)] = ∂za (z) for all α ∈ S. (3) The ﬁelds aα (z) are mutually local (4) V is spanned by the vectors α1 αm a ···a |0i, ji < 0 ( j1) ( jm)

Then these structures together with the vertex operation 1 1 α1 αm − j1−1 α1 − jm−1 αm Y(a( j ) ···a( j )|0i,z) = ··· : ∂z a (z)···∂z a (z) : 1 m (− j1 − 1)! (− jm − 1)! give rise to a unique vertex algebra structure on V satisfying (1)-(4) above and such that Y(aα ,z) = aα (z).

Assume that we have a r-vertex algebra satisfying the hypothesis of Theorem [?]. First we contend that α1 αm the vectors a ···a |0i , ji < 0 generate V as a vector space. Indeed, ( j1) ( jm) Lemma 5.4. Let a ∈ V be primitive and b ∈ V. For k ≥ 0, we have that (k) T (a) · b = a(−k−1)b.

Proof. For k = 0, we have that a · b = a(−1)b since a is primtive and V is a r-vertex algebra. The general case follows by induction using the following two facts: ﬁrst, if a is primtive, then so is T (k)(a); and second, d for any vertex algebra Y(Ta,z) = dzY(a,z).

This shows that any r-vertex algebra satisfying the hypothesis of Theorem 5.2, it will also satisfy the hypothesis of the Reconstruction theorem. Therefore, in order to show uniqueness, we just need to verify that the ﬁelds Y(aα ,z) are uniquely speciﬁed by the given information. So, let us suppose that (V,|0i,T,Y) and (V,|0i,T,Ye) are two b-vertex algebra structures satisfying the hypotheses of the theorem. We may as well assume that (V,|0i,T,Y) is the vertex algebra constructed in Theorem 5.2. Y a z a z−n−1 Y a z a z−n−1 Write e( , ) = ∑n∈Z e(n) and ( , ) = ∑n∈Z (n) . Then the lemma above allows us to conclude that Ye(a,z)+ = Y(a,z)+.

To show that Ye(a,z)−v = Y(a,z)−v, we ﬁrst check the result for v a primitive element of minimal positive degree d. Note than since V is primitively generated, the minimal degree of a primitive element is also the minimal positive degree of any element of V. By assumption, a(0)v = 0, so let us focus on a(n)v for n > 0. Each a(n) has strictly negative degree as an operator on V, so by our assumption on v we conclude that a(n)v = 0 unless n = d, in which case a(n)v = λ|0i. Furthermore, we can determine this value of λ from the 2-point functions: r(a,b) = (h0|,Y(a,z)b) = (ε,λ|0i)z−d−1 = λz−d−1. Hence for a primitive and v primitive of minimal degree Y(a,z)v = Ye(a,z)v. Now, we observe the following simple,

Lemma 5.5. Suppose Y(a,z)b = Ye(a,z)b, then Y(Ta,z)b = Ye(a,z)b and Y(b,z)a = Ye(b,z)a

9 Since V is T-generated from a set of (minimal degree) primitives, we see that Y(a,z)b is determined for all a,b ∈ V primitive. So in particular, a(n)b is determined for all a,b ∈ V primitive. It is easy to compute these to be:

• a(0)b = 0.

• for n > 0, a(n)b = 0 if n 6= deg(b)

• for n > 0, a(n)b = Coe f f (rz,0(a,b)) otherwise.

To ﬁnish the proof of the theorem, we need to understand the action of Y(a,z) on products of primitive elements. This is achieved through the following,

Proposition 5.6. Let Ye be a b-vertex algebra, and let a,v be a primitive. Then

Ye(a,z)−(v · w) = Ye(a,z)−v · w + v ·Ye(a,z)−w

Proof of Proposition. It is enough to show that for n ≥ 0

ae(n)[v · w] = ae(n)[v] · w + w · ae(n)[w]. This will follow essentially from the well known Borcherds commutator formulas, a special case of which is, n m [a(m),v(−1)] = ∑ (a( j)v)(m−1− j). j=0 j

But since for a and v primitive, a( j)v is always a scalar, so the only way (a( j)v)(m−1− j) can be a nonzero operator is when m = j. So, the formula reduces to

[a(m),v(−1)]w = (a(m)b)(−1)w.

Expanding this formula out and recalling that for x ∈ V primitive x(−1)y = x · y, we see that the claim is veriﬁed.

From the Proposition and the fact that V is T-generated by the set aα , the theorem is proven.

6. EXAMPLES

In this section, we show how the Heisenberg and Lattice Vertex algebras may be given the structure of a r-vertex algebra.

6.1. Heisenberg Vertex Algebras. Let h be an l-dimensional complex vector space equipped with a non- − denegerate, symmetric, bilinear form (·,·). The Heisenberg Lie algebra is defined to be hˆ = h ⊗ C[t,t 1] ⊕ m n n CK with relations [K,h] = 0 and [a ⊗t ,b ⊗t ] = (a,b)mδm,−nK. We will denote a ⊗t ,a ∈ h by a(n). For each λ in the dual space h∗, we may construct the Fock space by the following induction: λ ˆ Π = U(h) ⊗U(h⊗C[t]⊕CK) C where C is the one dimensional space on which h ⊗tC[t] acts as zero, K acts as the identity, and h ⊗t0 acts via the λ. As vector spaces, λ −1 −1 Π = Sym(h ⊗t C[t ]). Our next aim is to describe two r-vertex algebra structures on the vector space Π := Π0, and show that they are identical. We refer to this as the Heinsenberg vertex algebra. First, we will need to define a grading on Π, which is achieved by setting deg1 = 0 and dega1(− j1) · a2(− j2)···ak(− jk) · 1 = j1 + ··· jk and taking the grading by degree. Now Π has an algebra structure by construction, and it is also equipped with a derivation T : Π → Π defined by T · 1 = 0 and T(a(− j) · 1) = ja(− j − 1). Additionally, Π has the

10 structure of a bialgebra with derivation if we take h(−1),h ∈ h to be primitive elements and ε(1) = 1. Also, we note that Π has a bilinear form (·,·) deﬁned by (1,1) = 1 and such that h(n) and h(−n) are adjoint. With these preliminaries, we can use Theorem 4.5 to construct an r-vertex algebra structure on Π with (h,g) rz,w(h(−1),g(−1)) = (z−w)2 . Next, we recall the usual decription of the Heisenberg vertex algebra. In addition to the above data, −n−1 the vertex operators are described as follows: for h ∈ h, set h(z) = ∑n∈Z h(n)z where h(n) acts as the operator h ⊗tn on the hˆ-module Π. Explicitly, K acts as the identity, h(n) acts by multiplication for n < 0, derivation deﬁned by h(n)a(−s) = nδn,s(h,a) for n > 0, and 0 for n = 0. Then set 1 j1−1 − jk−1 Y(a1( j1)···ak( jk),z) = : ∂z a1(z)···∂z ak(z). (− j1 − 1)!···(− jk − 1)! Using Theorem 5.2, we easily conclude that these two vertex algebras are equal.

6.2. Lattice Vertex Algebras. Next, we consider the vertex algebra associated to a positive-definite, even lattice L. Suppose L is a free abelian group of rank l together with a positive, definite, integral bilinear form (·|·) : L ⊗ L → Z such that (α|α) is even for α ∈ L. Let h = C ⊗Z L equipped with the natural C-linear extension of the form (·|·) from L. Construct Π as above associated to h, and set VL = Π ⊗ C[L] where C[L] is the group algebra of L with basis eα ,α ∈ L. We will often write eα for 1 ⊗ eα and e0 = 1. α Since Π and C[L] are bialgebras, VL also has a bialgebra structure with gradation given by deg(e ) = 1 n 2 (α|α) and deg(h ⊗ t ) = n as before for h ∈ h. There is also a derivation T acting on VL by T · 1 = α α 0, T(h(−n) ⊗ 1) = nh(−n − 1) ⊗ 1, and T(1 ⊗ e ) = α(−1) ⊗ e . With this data VL is a bialgebra with α β derivation. VL is also equipped with a bilinear form defined by (1 ⊗ 1,1 ⊗ 1) = 1 and (h ⊗ e ,g ⊗ e ) = (h,h0)(α|β).

We proceed to give two descriptions of r-vertex algebra structures on VL, and then show they are iso- morphic. This vertex algebra is none other than the Lattice vertex algebra (of an even lattice). First, there is the usual vertex algebra constructed with vertex operators specified as follows: Let π1 be the rep- ˆ ˆ resentation of h on Π defined above. Also define a representation π2 of h on C[L] by setting π2(K) = 0, α α π2(h(n))e = δn,0(α,h)e for h ∈ H,α ∈ L,n ∈ Z. ˆ −n−1 Then VL becomes a h module by π = π1 ⊗ 1 + 1 ⊗ π2. Then for h ∈ h, set h(z) = ∑n∈Z h(n)z where n h(n) acts as the operator h ⊗t on the hˆ-module VL. Then, set 1 α j1−1 − jk−1 Y(a1( j1)···ak( jk) ⊗ e ,z) = : ∂z a1(z)···∂z ak(z)Γα (z) (− j1 − 1)!···(− jk − 1)! where z− j z− j α α −∑ j<0 α(− j) −∑ j>0 α(− j) Γα (z) = e z e j e j where zα (a ⊗ eβ ) = z(α,β)a ⊗ eβ and eα (a ⊗ eβ ) = ε(α,β)a ⊗ eα+β , and ε : L × L → ±1 is a 2-cocycle satisfying ε(α,β)ε(β,α) = (−1)(α|β)+(α|α)(β|β).

Secondly, we take take the r-vertex algebra deﬁned by Theorem 4.5 using the bicharacter r : VL ⊗VL → [(z − w)−1] r(eα ,eβ ) = ε(α,β) . C uniquely speciﬁed by (z−w)(α,β)

In order to show that the two descriptions of VL coincide, we would like to use Theorem 5.2, noting that they have the same two-point functions. Unfortunately, that theorem is not applicable since VL is not primitively generated (eα is group like). But, actually the proof of Theorem ? can be modified in this case as follows: Denote by (VL,1,T,Ye) and (VL,1,T,Y) the first and second vertex algebra descriptions above. VL is T-generated from C[L], so by the Reconstruction Theorem, we just need to see that the fields Y(eα ,z) = Ye(eα ,z). Step 1: Given h ∈ h, let us show that the fields Y(h(−1),z) = Ye(h(−1),z). The proof of Theorem 5.2 shows that Y(h(−1),z)+ = Ye(h(−1),z)+, so what remains is to show that Y(h(−1),z)− = Ye(h(−1),z)−.

11 On Π these two agree by the same reasoning as in the proof of Theorem 5.2. Furthermore, on C[L], we contend that (h,α)eα Y(h(−1),z) eα = . − z Indeed, since the 2-point functions are (h,α)eα (ε,Y(h(−1),z)α ) = r(h(−1),eα ) = , e z we know that α α h(−1)( )e λe Y(h(−1),z)eα = 0 = , e z z since ae(0) is a scalar on group-like elements for a ∈ VL primitive. Clearly λ is speciﬁed by r and so we have shown that Ye = Y on C[L] as well. Now, use Proposition [?] to conclude that Ye = Y on all of VL. α n α α n α Step 2: [h(−1)(n),Y(e ,z)] = (α|h)z Y(e ,z) and [h(−1)(n),Ye(e ,z)] = (α|h)z Ye(e ,z). This follows easily from Borcherds’s commutator formula and the fact that Y(h(−1),z) = Ye(h(−1),z). Note that in particular [h(0),Y(eα ,z)] = (α|h)Y(eα ,z) and similarly [h(0),Y(eα ,z)] = (α|h)Ye(eα ,z). This means that

α β β+α Y(e ,z) : Π ⊗ C[e ] → Π ⊗ C[e ], and similarly for Ye(eα ,z). α 1 Step 3: Since e has degree 2 (α,α), we may write

α α −n− (α,α) Y(e ,z) = ∑ e [n]z 2 n∈Z and α α −n− (α,α) Y(e ,z) = ∑ e˜ [n]z 2 n∈Z where eα [n] ande ˜α [n] have degree −n. Step 4: We show that Y(eα ,z)eβ = Ye(eα ,z)eβ for α,β ∈ L. Indeed, by degree considerations, we have that (α,α) eα [s]eβ = 0 for s > − − (β,α) 2 and (α,α) eα [s]eβ = c eα+β for s = − − (β,α). α,β 2 α Similarly fore ˜ [s], and examining 2-point functions we see that the cα,β are completely determined. Now, using induction and the fact that d Y(eα ,z) = Y(Teα ,z) =: Y(α(−1),z)Y(eα ,z) dz and d Y(eα ,z) = Y(Teα ,z) =: Y(α(−1),z)Y(eα ,z) dz e e e e we see that Y(eα ,z)eβ = Ye(eα ,z)eβ . α β α β Step 5: Using Step 2, we may compute Y(e ,z)h1(−n1)···hk(−nk)e and Ye(e ,z)h1(−n1)···hk(−nk)e and see that they are equal by Step 1 and 4.

12 7. SOME QUESTIONS

REFERENCES

[Bo] R. Borcherds, Quantum vertex algebras. Taniguchi Conference on Mathematics Nara ’98, 51–74, Adv. Stud. Pure Math., 31, Math. Soc. Japan, Tokyo, 2001. [Na] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces. Annals of Math (2) 145 (1997), no.2, pp. 379-388. [FB] E. Frenkel, D. Ben-Zvi, Vertex algebras and algebraic curves. Mathematical Surveys and Monographs, 88. American Mathe- matical Society, Providence, RI, 2001.